FOM: Cantor'sTheorem & Paradoxes & Continuum Hypothesis
kanovei at wmwap1.math.uni-wuppertal.de
Mon Feb 12 14:20:45 EST 2001
>From: Neil Tennant <neilt at mercutio.cohums.ohio-state.edu>
>Cantor's theorem does not depend on the assumption that
the power set of the continuum exists.
Once again, the proof has two important issues,
1) technically - the diagonal method (which in fact was used,
perhaps, earlier by Du Bois Reymond in "scaling" sequences,
2) foundationally - the postulate that all elements of P(N)
(or P(X),for "arbitrary" X - which was not much meaningful
for a XIX century mathematician)
are "already given" and no new ones can appear in the course
of the proof.
The rest is a couple of lines of ordinary transformations.
Something similar to 2) appears in many paradoxes, say those
of Liar and Barber in the form of a mess between "has been"
and "has been and will always be".
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